Canonical quasi-geodesic mappings of special pseudo-Riemannian spaces

Abstract

The present paper continues the study of quasi-geodesic mappings f:(V_n, g_ij, F_i^h) → (V'_n,g'_ij, F_i^h) of pseudo-Riemannian spaces V_n, V'_n with a generalized-recurrent structure F_i^h of parabolic type. By a generalized recurrent structure of parabolic type on V_n we mean an almost Hermitian affinor structure of parabolic type for which the covariant derivative of the structural affinor F_i^h satisfies the condition F_(i,j)^h=q_(i F_j)^h.

In the previous paper by the authors [Proc. Intern. Geom. Center, 13:3 (2020) 18-32] it was proved that the class of pseudo-Riemannian spaces with generalized-recurrent structure of parabolic type is closed with respect to the considered mappings and the generalized recurrence vectors in (V_n, g_ij,F_i^h) and (V'_n, g'_ij, F_i^h) may be distinct. In this article, it is assumed that the mapping f preserves the generalized recurrence vector q_i.

We construct geometric objects that are invariant under the quasi-geodesic mapping of generalized-recurrent spaces of parabolic type and recurrent-parabolic spaces. A number of conditions are given on these objects, which lead to the fact that a generalized-recurrent space of parabolic type admits a parabolic K-structure, and a recurrent-parabolic space admits a Kählerian structure of parabolic type.

We study special types of these mappings that preserve some tensors of an intrinsic nature.